The maximally symmetric nonflat path structure has dimension of the symmetry algebra \(n^2+5\) , see [1]}.
Uniqueness of such a structure has been recently established in [2]}.
For \(n+1=3\)  this structure was discussed in [3]} in relation to self-dual gravity,
the corresponding spacetimeThis Plebanski type metric has coordinate expression
\(g=dx\,dw+dy\,dz-y^2dw^2\) . is Ricci flat of Petrov type N.
The ODE system generating this metric via the twistor correspondence is
\(\ddot{y}^1=(\dot{y}^2)^3,\ \ddot{y}^2=0,\ \dots ,\ \ddot{y}^n=0.\) 
